Approximation Theory assists in the science and math of reconstructing signals. The general principles of Approximation Theory are concerned with how functions may be approximated with other, simpler functions, and with characterizing in a quantitative way, the errors introduced thereby. With signal reconstruction, the fundamental problem is to reconstruct a signal that is a vector (or series) of Real Numbers from linear measurements with respect to a dictionary for vectors of Real Numbers. Currently through the use of Approximation Theory, there are vast computations involved in order to reconstruct a signal.
Current developments in Approximation Theory have evolved a new field within it that allows for fewer computations in order to reconstruct a signal. This new field is called Compressed Sensing. With Compressed Sensing, reconstruction of a signal may be done with very few linear measurements over a modified dictionary if the information of the signal is concentrated in coefficients over an orthonormal basis. These results have reconstruction error on any given signal that is optimal with respect to a broad class of signals. The field of Compressed Sensing allows for an innovative approach to allow a much smaller number of calculations than the signal size to reconstruct a signal but there is yet no method developed to accomplish this using an algorithmic approach. An algorithmic approach allows showing that Compressed Sensing results resonate with prior work in Group Testing, Learning theory, and Streaming algorithms. Current technology needs a new method that allows these new algorithms to present the most general results for Compressed Sensing with an approximation on every signal, faster algorithms for the reconstruction as well as succinct transformations of the dictionary to the modified dictionary.